Sampling & Probability

Sampling

Research Question

Every research project aims to answer a research question (or multiple questions).

Do ECU students who exercise regularly have a higher GPA?

Population

Each research question aims to examine a population.

Population for this research question is ECU students.

Survey Questions

  • Do you exercise at least once a week?
  • What is your GPA?

Sampling

  • It is impossible to study the whole population related to a research question.

  • A sample \(n\) is a subset of the population \(N\).

  • The Goal: Select a representative sample to generalize to the broader population.

What is representative?

Convenience Sampling

  • Sample is easy to access.


  • Example:
    • Stand in front of Joyner Library.
    • Give the survey to 100 ECU students.


  • Issue:
    • Will introduce sampling bias

PSA x2

Data quality matters more than data quantity


Many anthropological studies (or similar) are convenience based.

Simple Random Sample

Every member of a population has an equal chance of being selected.

  • Examples:
    • Reach out to the registrar for student emails
    • Randomly select 100 students
    • Email students the survey

Simple Random Sampling in R

sample(1:100, 3, replace = FALSE)
[1] 38 53 14

To Generalize:

sample(x = 1:N, size = n, replace = FALSE)

Systematic Sampling

Similar to a simple random sample BUT intervals are chosen at regular intervals.

# 1. Create a population (e.g., a vector of 1 to 1000)
population <- 1:1000

# 2. Define the desired sample size
sample_size <- 100

# 3. Calculate the sampling interval (k)
N <- length(population) # Population size
k <- N / sample_size
# If k is not an integer, you might use ceiling(N/n) and adjust the logic

# 4. Choose a random starting point (r) between 1 and k
set.seed(123) # Optional: for reproducible results
start_point <- sample(1:k, 1)

# 5. Select every k-th element starting from the random start point
systematic_sample_indices <- seq(from = start_point, to = N, by = k)
systematic_sample <- population[systematic_sample_indices]

# 6. View the first few elements and the dimension of the sample
head(systematic_sample)
[1]  3 13 23 33 43 53
length(systematic_sample) # Should be the desired sample size (100)
[1] 100

Subgroups

Stratified Random Sampling in R

library(dplyr)

# Sample data
set.seed(123) # For reproducibility
data <- data.frame(
  ID = 1:100,
  Gender = sample(c("Male", "Female"), 100, replace = TRUE),
  Income = rnorm(100, mean = 50000, sd = 10000)
)
# Stratified sampling with sample_n()
sampled_data_n <- data %>%
  group_by(Gender) %>%
  sample_n(10)

# View the sampled data
# sampled_data_n %>% count(Gender)

Single Stage Cluster Sampling in R

set.seed(123)

population <- data.frame(
  Supermarket = paste("Supermarket", 1:1000, sep = "_"),
  CustomerSatisfaction = rnorm(1000, mean = 75, sd = 10)
)

selected_supermarkets <- sample(population$Supermarket, size = 10, replace = FALSE)

sampled_data <- population[population$Supermarket %in% selected_supermarkets, ]

head(sampled_data)
        Supermarket CustomerSatisfaction
203 Supermarket_203             72.34855
225 Supermarket_225             71.36343
255 Supermarket_255             90.98509
354 Supermarket_354             76.16637
457 Supermarket_457             86.10277
554 Supermarket_554             77.49825

2-Stage Cluster Sampling

set.seed(123)

region <- data.frame(
  Neighborhood = paste("Neighborhood", 1:500, sep = "_"),
  AverageIncome = rnorm(500, mean = 50000, sd = 10000)
)

households <- data.frame(
  Neighborhood = rep(sample(region$Neighborhood, size = 500, replace = TRUE), each = 20),
  HouseholdID = rep(1:20, times = 500),
  EmploymentStatus = sample(c("Employed", "Unemployed"), size = 10000, replace = TRUE)
)

selected_neighborhoods <- sample(region$Neighborhood, size = 5, replace = FALSE)

sampled_households <- households[households$Neighborhood %in% selected_neighborhoods, ]

head(sampled_households)
         Neighborhood HouseholdID EmploymentStatus
1981 Neighborhood_302           1       Unemployed
1982 Neighborhood_302           2         Employed
1983 Neighborhood_302           3         Employed
1984 Neighborhood_302           4         Employed
1985 Neighborhood_302           5       Unemployed
1986 Neighborhood_302           6       Unemployed

Multi-Stage Cluster Sampling

set.seed(123)
states <- data.frame(
  State = paste("State", 1:50, sep = "_"),
  Population = sample(1000000:5000000, 50, replace = TRUE)
)
counties <- data.frame(
  State = rep(sample(states$State, size = 50, replace = TRUE), each = 20),
  County = rep(paste("County", 1:20, sep = "_"), times = 50),
  VaccinationRate = rnorm(1000, mean = 70, sd = 5)
)
selected_states <- sample(states$State, size = 3, replace = FALSE)
selected_counties <- sample(counties$County[counties$State %in% selected_states], size = 5, replace = FALSE)
sampled_vaccination_centers <- counties[counties$County %in% selected_counties, ]
head(sampled_vaccination_centers)
      State    County VaccinationRate
8  State_32  County_8        70.37428
11 State_32 County_11        66.86024
13 State_32 County_13        70.81309
15 State_32 County_15        67.68222
19 State_32 County_19        70.91839
28 State_46  County_8        68.84869

# 1. Define Population Distribution (e.g., skewed population)
set.seed(123)
population <- rgamma(100000, shape = 2, scale = 2)

# 2. Take a Sample Distribution (e.g., 100 individuals)
sample_data <- sample(population, 100)

From Sampling to Probability

How do we infer future events or population characteristics?

Random Process

In a random process there is more than one possible outcome.

  • The deterministic prediction of the outcome is difficult to impossible.
sample(x = c("H", "T"), size = 10, replace = T)
 [1] "T" "T" "T" "T" "H" "T" "H" "T" "H" "H"
sample(x = c(1:6), size = 10, replace = T)
 [1] 3 6 5 6 3 5 4 1 5 2

Random vs. Deterministic Processes

Sample Space

The set of all possible outcomes of a random process.

Event

An event is a subset of the sample space.

  • Examples with a 6-sided die:

    • Let A represent the event that a single roll die results in an even number.
      • A = {2, 4, 6}
    • Let B represent the event that a single roll die results in an odd number.
      • B = {1, 3, 5}
    • Let C represent the event that a single roll die results in a prime number.
      • C = {2, 3, 5}

Complement of an Event

The set of all outcomes in the sample space that are not in the event itself.

  • Example:

    • Let C represent the event that a single roll die results in a prime number.
      • C = {2, 3, 5}
    • Notation: \(C^C\)
      • C complement
    • \(C^C\) = {1, 4, 6}

Mutually Exclusive or Disjoint Events

    • Let A represent the event that a single roll die results in an even number.
      • A = {2, 4, 6}
    • Let B represent the event that a single roll die results in an odd number.
      • B = {1, 3, 5}
    • Let C represent the event that a single roll die results in a prime number.
      • C = {2, 3, 5}

Events \(A\) and \(B\) are mutually exclusive because an outcome cannot be both even + odd.

Events \(A\) and \(C\) are not mutually exclusive because the outcome 2 is both even + prime.

Events & Set Notation



Description Notation Reading Elements
Union \(A \cup C\) A or C {2, 3, 4, 5, 6}
Intersection \(A \cap C\) A and C {2} 

set.seed(1)
OBV <- 1:10
Dist1 <- NULL
Dist9 <- NULL
Dist16 <- NULL
Dist25 <- NULL
Dist36 <- NULL
count = 100
while(count > 0){Dist1 <- c(Dist1,sample(OBV, 1, replace = TRUE)); count <- count - 1}
count = 100
while(count > 0){Dist9 <- c(Dist9,mean(sample(OBV, 9,replace = TRUE) ) ); count <- count - 1}
count = 100
while(count > 0){Dist16 <- c(Dist16,mean(sample(OBV, 16,replace = TRUE) ) ); count <- count - 1}
count = 100
while(count > 0){Dist25 <- c(Dist25,mean(sample(OBV, 25,replace = TRUE) ) ); count <- count - 1}
count = 100
while(count > 0){Dist36 <- c(Dist36,mean(sample(OBV, 36,replace = TRUE) ) ); count <- count - 1}
Dist.df <- data.frame(Size = factor(rep(c(1,9,16,25,36), each=100)), Sample_Means = c(Dist1, Dist9, Dist16, Dist25, Dist36) )
ggplot(Dist.df, aes(Sample_Means, fill = Size)) + geom_histogram() + facet_grid(. ~ Size)